It can be noticed easily that the Laplacians \(L_4^{^{\prime }}\)  and \(L_4^{^{\prime \prime }}\)  have the common algebraic connectivity as \(\lambda _2=4\)  and the common Fiedler vector as \(\mathbf {v}_2 = \mathbf {v}_F = [-1, -1, 3, -1]^T\) , and these matrices satisfy the condition (REF ). The corresponding graphs with same connectivity, \(G(\lbrace  1,2,3,4^{^{\prime }} \rbrace )\)  and \((\lbrace  1,2,3,4^{^{\prime \prime }} \rbrace )\)  , are shown in Figure REF . In this case, the movement of an agent doesn't influence the global connectivity which is counter intuitive [1]}. In fact, these topologies offer freedom of moving an agent in a network without compromising its connectivity.
<FIGURE>